Nonparametric tests. Nonparametric tests and ANOVAs: What you need to know. Quick Reference Summary: Sign Test. ( ) n(x P = 2 * Pr[x!

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1 Nonparametric tests and ANOVAs: What you need to know Nonparametric tests Nonparametric tests are usually based on ranks There are nonparametric versions of most parametric tests arametric One-sample and aired t-test Two-sample t-test Nonparametric Sign test Mann-Whitney U-test Quick Reference Summary: Sign Test What is it for? A non-parametric test to compare the medians of a group to some constant What does it assume? Random samples Formula: Identical to a binomial test with p o = 0.. Uses the number of subjects with values greater than and less than a hypothesized median as the test statistic. " (x) = probability of a total of x successes p = probability of success in each trial n = total number of trials (x) = n% $ ' p x 1( p # x& ( ) n(x = * r[x!x]

2 Sample Test statistic x = number of values greater than m o Reject H o Sign test compare How unusual is this test statistic? < 0.0 > 0.0 Null hypothesis Median = m o Null distribution Binomial n, 0. Fail to reject H o Quick Reference Summary: Mann-Whitney U Test What is it for? A non-parametric test to compare the central tendencies of two groups What does it assume? Random samples Test statistic: U Distribution under H o : U distribution, with sample sizes n 1 and n Formulae: ( ) U 1 = n 1 n + n n U = n 1 n " U 1 " R 1 n 1 = sample size of group 1 n = sample size of group R 1 = sum of ranks of group 1 Use the larger of U1 or U for a two-tailed test Sample Test statistic U 1 or U (use the largest) Mann-Whitney U test compare How unusual is this test statistic? < 0.0 > 0.0 Null hypothesis The two groups Have the same median Null distribution U with n 1, n Mann-Whitney U test Large-sample approximation: Z = U " n 1 n n 1 n n 1 + n +1 ( ) / Use this when n 1 & n are both > 10 Compare to the standard normal distribution Reject H o Fail to reject H o

3 Mann-Whitney U Test If you have ties: Rank them anyway, pretending they were slightly different Find the average of the ranks for the identical values, and give them all that rank Carry on as if all the whole-number ranks have been used up TIES

4 Rank A TIES Rank them anyway, pretending they were slightly different 1 7 Rank A 1 7 Find the average of the ranks for the identical values, and give them all that rank Rank A 1 7 Average = 1. Average =

5 Rank A Rank Rank A Rank These can now be used for the Mann-Whitney U test Benefits and Costs of Nonparametric Tests Main benefit: Make fewer assumptions about your data E.g. only assume random sample Main cost: Reduce statistical power Increased chance of Type II error When Should I Use Nonparametric Tests? When you have reason to suspect the assumptions of your test are violated Non-normal distribution No transformation makes the distribution normal Different variances for two groups

6 Quick Reference Summary: ANOVA (analysis of variance) What is it for? Testing the difference among k means simultaneously What does it assume? The variable is normally distributed with equal standard deviations (and variances) in all k populations; each sample is a random sample Test statistic: F Distribution under H o : F distribution with k-1 and N-k degrees of freedom Quick Reference Summary: ANOVA (analysis of variance) Formulae: = group F = = k "1 (Y i "Y) Y i Y = mean of group i = overall mean error N " k = # s i n i = size of sample i N = total sample size k Samples Test statistic F = ANOVA compare How unusual is this test statistic? < 0.0 > 0.0 Null hypothesis All groups have the same mean Null distribution F with k-1, N-k Quick Reference Summary: ANOVA (analysis of variance) Formulae: F = There MS are error a LOT of = equations = here, group and k "1 this is the error N " k simplest possible SS = s group = n i (Y i "Y) ANOVA i # # Reject H o Fail to reject H o Y i Y = mean of group i = overall mean n i = size of sample i N = total sample size

7 F = (Y i "Y) group = k-1 = # s i error = N-k = = group k "1 error N " k F = = group = k "1 error N " k (Y i "Y) = # s i F-ratio ANOVA Tables (Y i "Y) group = k-1 = = group k "1 F = = # s i error = N-k error N " k

8 ANOVA Tables ANOVA Tables (Y i "Y) (Y i "Y) k-1 = # s i = # s i N-k + + N-1 ANOVA Tables ANOVA Tables (Y i "Y) k-1 = group (Y i "Y) k-1 = group F = = # s i N-k error = # s i N-k error + N-1 + N-1

9 ANOVA Tables ANOVA Table: (Y i "Y) k-1 = group F = * 7. = # s i N-k error N-1 ANOVA Table: Additions to ANOVA R value: how much variance is explained? Comparisons of groups: planned and unplanned Fixed vs. random effects 1. 1 Repeatability

10 Two-Factor ANOVA Often we manipulate more than one thing at a time Multiple categorical explanitory variables : sex and nationality Two-factor ANOVA Don t worry about the equations for this Use an ANOVA table Two-factor ANOVA Two-factor ANOVA Table Testing three things: Square 1. s don t differ among treatment 1. s don t differ among treatment. There is no interaction between the two treatments 1 1 * SS 1 SS SS 1* k 1-1 k - 1 (k 1-1)*(k - 1) SS 1 k 1-1 SS k - 1 SS 1* (k 1-1)*(k - 1) MS 1 MSE MS MSE MS 1* MSE XXX XXX SS total N-1

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